In the present paper we find a bijection between the set of small covers over an –cube and the set of acyclic digraphs with labeled nodes. Using this, we give formulas of the number of small covers over an –cube (generally, a product of simplices) up to Davis–Januszkiewicz equivalence classes and –equivariant homeomorphism classes. Moreover we prove that the number of acyclic digraphs with unlabeled nodes is an upper bound of the number of small covers over an –cube up to homeomorphism.
"The number of small covers over cubes." Algebr. Geom. Topol. 8 (4) 2391 - 2399, 2008. https://doi.org/10.2140/agt.2008.8.2391